General Relativity and Quantum Cosmology

Title:
The Scalar Curvature of a Causal Set

Abstract: A one parameter family of retarded linear operators on scalar fields on
causal sets is introduced. When the causal set is well-approximated by 4
dimensional Minkowski spacetime, the operators are Lorentz invariant but
nonlocal, are parametrised by the scale of the nonlocality and approximate the
continuum scalar D'Alembertian, $\Box$, when acting on fields that vary slowly
on the nonlocality scale. The same operators can be applied to scalar fields on
causal sets which are well-approximated by curved spacetimes in which case they
approximate $\Box - {{1/2}}R$ where $R$ is the Ricci scalar curvature. This can
used to define an approximately local action functional for causal sets.

Comments:

Typo in definition of equation (3) and definition of n(x,y) corrected. Note: published version still contains typo